There is a thread on this board where someone has done that and more or less got it to work (some searching should find it). It will probably work out reasonably well with two identical batteries (i.e. two new ones).

I would recommend against adding a secondary battery with a different usage history as the one already present or replacing an individual battery of the pair; there is control circuitry involved to handle the Lithium Ion battery properly over its lifetime. As you simply tie them together, I don't think that will work out if the batteries don't respond the same at all times; you could wear at least one out quickly and even risk setting it on fire.

Yes, on closer examination I saw the control circuitry on the battery itself, which can complicate things a lot. I've been thinking of other possible setups like a 3-way switch to turn the device off, or alternate between batteries 1 and 2, but after all, what's the point in trying to juggle with soldering stuff to increase the battery capacity if you can just buy a double capacity battery that fits in (almost)? See this very neat 3.7V, 2400mAh Li-Po battery in eBay for under 15 bucks. It's 110mm(L)*40mm(W)*5.2mm(H) in size, so it should fit snugly except for length; it even has a 2000mAh cousin that is 3.9mm thick if 5.2mm is a bit too much, and I'm sure intermediate options will be available if necessary.

The extra length should be easy to accommodate with some plastic ribbing dremeling in a worst case scenario, so I'm definitely buying this battery and I'll do some experiments (turning sleep mode off and see how long these batteries last/take to charge compared to the original) on another DR1000 I got supercheap for parts (the screen is a mess but works perfectly otherwise), and I'll report back next month.